As geometries continue to shrink, manufacturers have increasingly turned to optical techniques to perform non-destructive inspection and analysis of semiconductor wafers. Techniques of this type, known generally as optical metrology, operate by illuminating a sample with electromagnetic radiation and then detecting and analyzing the reflected energy. Scatterometry is a specific type of optical metrology that is used when the structural geometry of a subject creates diffraction (optical scattering) of the incoming probe beam. Scatterometry systems analyze diffraction to deduce details of the structures that cause the diffraction to occur. In general, optical metrology measures physical dimensions and/or material properties of structures on a wafer from the optical response of the wafer, that is, how the wafer changes light that illuminates it.
Various optical techniques have been used to perform optical scatterometry. These include broadband spectroscopy (U.S. Pat. Nos. 5,607,800; 5,867,276 and 5,963,329), spectral ellipsometry (U.S. Pat. No. 5,739,909) single-wavelength optical scattering (U.S. Pat. No. 5,889,593), and spectral and single-wavelength beam profile reflectance and beam profile ellipsometry (U.S. Pat. No. 6,429,943). Scatterometry, in these cases generally refers to optical responses in the form of diffraction orders produced by period structures, that is, gratings on the wafer. In addition it may be possible to employ any of these measurement technologies, e.g., single-wavelength laser BPR or BPE, to obtain critical dimension (CD) measurements on non periodic structures, such as isolated lines or isolated vias and mesas. The above cited patents and patent applications, along with PCT Application WO03/009063, U.S. application Ser. No. 2002/0158193, U.S. application Ser. No. 10/243245, filed Sep. 13, 2002, U.S. application Ser. No. 2001/0051856 A1, PCT Application WO 01/55669 and PCT Application WO 01/97280 are all incorporated herein by reference.
Metrology uses three classes of parameters and their relationships. Structures on the wafer have physical parameters, e.g., the thickness of a layer, the widths of a line structure at various heights (measured generally perpendicular to a face of the wafer, or the complex optical index of a material. Most scatterometry measurements are performed over a range of independent parameters. Examples independent parameters are wavelength (for spectroscopic systems) or angle of propagation. Much of the following is written as if wavelength is the independent parameter. The goal of metrology is to relate measurements to a model of what is on the wafer, where the model has parameters that in some sense reflect the physical parameters on the wafer. In some cases, model parameters represent exactly the physical parameters in question, or they may be related through some mathematical transformation, e.g., the physical widths of adjacent periodic lines and spaces may be modeled as a period and ratio.
Most scatterometry systems use a modeling approach to transform empirical data into tangible results. For this type of approach, a theoretical model is defined for each subject that will be analyzed. The theoretical model predicts the output (reflected) electromagnetic field that is generated when an incident field is applied to the subject. The theoretical model is parameterized and each parameter corresponds to a physical characteristic of the subject such as line width or layer thickness. A profile is a collection of line widths at various heights, and is thus a collection of parameters. A regression is performed in which the parameters are repeatedly perturbed and the model is repeatedly evaluated to minimize the differences between the modeled results and results that are empirically obtained. The differences are typically calculated over the range of the independent parameters, and then an average difference, such as a squared sum, is calculated as a single difference. Various norms or other techniques are suitable for collapsing the multiple differences into a single working difference. When the minimization reaches some stopping criterion, it is assumed that the model and its associated parameters accurately reflect the subject being analyzed. One such stopping criterion is that the difference reaches some predetermined level, e.g., that a goodness-of-fit criterion is exceeded. Another criterion is reached when the reduction of the difference from repeat to repeat becomes sufficiently small. Others are possible.
Evaluation of theoretical scatterometry models is a complex task, even for relatively simple subject. As subjects become more complex, e.g., having more parameters, the calculations can become extremely time-consuming. Even with high-speed processors, real-time evaluation of these calculations can be difficult. This is problematic in semiconductor manufacturing where it is often imperative to detect processes that are not operating correctly. As the semiconductor industry moves towards integrated metrology solutions (i.e., where metrology hardware is integrated directly with process hardware) the need for rapid evaluation becomes even more acute.
A number of approaches have been developed to overcome the calculation bottleneck associated with the analysis of scatterometry results. Many of these approaches have involved techniques for improving calculation throughput, such as parallel processing techniques. For example, co-pending application Ser. No. 09/906,290 filed Jul. 16, 2001 describes a system in which a master processor distributes scatterometry calculations among a group of slave processors. This can be done by as a function of wavelength, so that each slave processor evaluates the theoretical model for selected wavelengths. The other slave processors will carry out the same calculations at different wavelengths. Assuming there are five processors (one master and four slaves) and fifty wavelengths, each processor will perform ten such calculations per iteration. Once complete, the master processor combines the separate calculation and performs the best fit comparison to the empirical results. Based on this fit, the master processor will modify the parameters of the model (e.g. changing the widths or layer thickness) and distribute the calculations for the modified model to the slave processors. This sequence is repeated until a good fit is achieved.
This distributed processing approach can also be used with multiple angle of incidence information. In this situation, the calculations at each of the different angles of incidence can be distributed to the slave processor. Techniques of this type are an effective method for reducing the time required for scatterometry calculations. At the same time, the speedup provided by parallel processing is strictly dependent on the availability (and associated cost) of multiple processors. Amdahl's law also limits the amount of speedup available by parallel processing since serial portions of the program are not improved. At the present time, neither cost nor ultimate speed improvement is a serious limitation for parallel processing techniques. As the complexity of the geometry increases, however it becomes increasingly possible that computational complexity will outstrip the use of parallel techniques alone.
Another approach to rapidly evaluating scatterometry measurements is to use pre-computed libraries of predicted measurements. This type of approach is discussed in PCT application WO 99/45340, published Sep. 10, 1999 as well as the references cited therein. In this approach, a library of expected results is constructed by repeatedly evaluating the theoretical model for range of different parameters. When empirical measurements are obtained, the library is searched to find the best fit.
The use of libraries speeds the analysis process by allowing theoretical results to be computed once and reused many times. Of course, libraries are necessarily limited in their resolution and can contain only a finite number of theoretical results. This means that there are many cases where empirical measurements do not have exact library matches. In these cases, the use of a library represents an undesirable choice between speed and computational accuracy.
To overcome this limitation, U.S. Patent Application Publication No. 20020038196A1 (incorporated in this document by reference) describes a database method of analysis of empirical scatterometry measurements. The database method is similar to the library approach in that it relies on a stored set of pre-computed “reflectance characteristics.” (The reflectance characteristics could be simulated reflectance signal spectra, although in the preferred embodiment they are complex reflectance coefficients from which the reflectance signal can be computed without resorting to complex electromagnetic simulations). In this case, however an interpolation method is used in combination with the database-stored characteristics, making it possible to achieve measurement resolution and accuracy much better than the database sampling resolution. Both the database size and computation time are consequently greatly reduced relative to library-based methods.
A critical element of the database interpolation method is the interpolation algorithm itself. In the application just cited (i.e., U.S. Patent Application Publication No.20020038196A1), two preferred algorithms are disclosed: multilinear and multicubic. Multilinear interpolation is very fast, but has poor accuracy. Multicubic interpolation is much more accurate, but can be slow, especially when many parameters are being simultaneously measured. In practice, selection of between multilinear and multicubic method is done on a pragmatic basis depending on the degree of accuracy and speed required. Often, this choice is totally acceptable. This is true, for example, when the number of parameters is relatively small. Still, it is clear that interpolation methods that provide increased speed and accuracy would be an improvement over current techniques.